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4. Groups of Permutations 1 4. Groups of permutations 1 4. Groups of permutations Consider a set of n distinguishable objects, fB1;B2;B3;:::;Bng. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given permutation operation is 1 2 3 : : : n ! , a1 a2 a3 : : : an where the numbers fa1; a2; a3; : : : ; ang are the numbers f1; 2; 3; : : : ; ng in some order. This operation can be interpreted in two different ways, as follows. Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set. Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical { for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD. Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant { the columns may be written in any order without affecting the result, provided each column is kept intact. The chronological order in which the objects are rearranged doesn't matter. 1 2 3 4 ! As an example, consider the action of the permutation 4 1 3 2 on the ordered set wxyz. Using interpretation 1, the first object, w, is moved to the 4th position; the second object, x, to the 1st position; the third object, y, to the 3rd position; the fourth object, z, to the 2nd position, producing the final order xzyw. In order to use interpretation 2, the objects in the Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 2 set must be relabeled | x1 = w; x2 = x; x3 = y; x4 = z | after which the label 1 is replaced by 4, the label 2 by 1, the label 3 by 3 and the label 4 by 2, producing the final order x4x1x3x2 = zwyx. Both interpretations lead to well-defined permutations of the symbols w; x; y; z, but give different results, so it is essential to specify the interpretation being used. Now consider the effect of multiplying permutations, where multiplication is defined as consecutive action of two permutations. (Recall the standard convention that the right-hand factor in a product acts first, followed by the left-hand factor.) On the result of the above example, act with the permu- 1 2 3 4 ! tation . Using interpretation 1, it acts on xzyw to produce 4 3 2 1 1 2 3 4 ! wyzx, which is produced from wxyz by the permutation . 1 4 2 3 1 2 3 4 ! 1 2 3 4 ! 1 2 3 4 ! This multiplication can be written = . 4 3 2 1 4 1 3 2 1 4 2 3 By reordering the columns in the left-hand factor of the product so that the top row reads the same as the bottom row of the right-hand factor, this can 4 1 3 2 ! 1 2 3 4 ! 1 2 3 4 ! be rewritten in the form = . 1 4 2 3 4 1 3 2 1 4 2 3 In this form, it is evident that multiplication can be carried out by \cancel- ing" the bottom row of the right-hand factor against the identical top row of the left-hand factor. The result has the top row of the right-hand factor and the bottom row of the left-hand factor. A moment's consideration will show that this is a universal feature (it simply says, for example, move the 1st object to the 4th position in the first step and the 4th object to the 1st position in the second step, leaving the 1st object in the 1st position overall, and so on), which makes multiplication of permutations trivial. The same process can be carried out using interpretation 2. Now the second permutation acts on zwyx (which is x4x1x3x2) to produce x1x4x2x3, or wzxy, which is also the result of acting on wxyz with the permutation 1 2 3 4 ! . The multiplication is expressed in permutation symbols in 1 4 2 3 exactly the same way as in interpretation 1, and the same rule of multiplica- tion by cancellation can be applied. [The two interpretations again lead to different final results, wyzx and wzxy, even though they are applied to the same symbolic multiplication.] The cancellation rule for evaluating products of permutations shows triv- ially that the multiplication is associative. There is an identity permutation Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 3 1 2 : : : n ! , and the inverse of any permutation is obtained by ex- 1 2 : : : n changing its top and bottom rows. So these transformations constitute a group, where multiplication is understood as consecutive action of the trans- formations. 1. A permutation of the set fBig, followed by another permutation, clearly produces a permutation of the original set, so the permutations are closed under multiplication. 2. The multiplication of permutations has been shown to be associative. (A permutation can be regarded as a mapping of the set of ordered n-tuples of integers, confirming that the multiplication is associative.) 3. The permutation which leaves the ordered set unchanged is clearly an identity. 4. Given a permutation which changes an initial ordering of the set fBig into a final ordering, the permutation which changes the final ordering back into the initial ordering is the inverse of the starting permutation. So the set of all permutations of n objects forms a group, called the symmetric group on n objects, denoted Sn. It has order n!. An alternative, more economical and very efficient notation for permu- tations is provided by cycles.A cycle is a sequence of up to n labels, (a1; a2; a3; : : : ; am)(m ≤ n) and represents a permutation symbol a a a : : : a a b : : : b ! 1 2 3 m−1 m 1 n−m , a2 a3 a4 : : : am a1 b1 : : : bn−m where fbig are all the n − m labels not included in the set faig. It permutes the labels faig cyclically and leaves the labels fbig unchanged. The number of labels it contains is the degree of the cycle. A cycle of degree m is referred to as an m-cycle. It is easily seen that two cycles with no label in common commute with one another, while two cycles with any labels in common will generally not commute with one another. The order of the entries in a cycle is significant, but since it is cyclic, the starting point is irrelevant: (a1; a2; : : : ; am) = (a2; : : : ; am; a1), etc., so an m-cycle can be written in m different but equivalent ways. For example, (a1; a2; a3; a4) = (a2; a3; a4; a1) = (a3; a4; a1; a2) = (a4; a1; a2; a3). The order of a cycle is equal to its degree | acting m times with an m- cycle restores all labels to their starting positions, i.e. is equal to the identity. Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 4 Any 1-cycle is equal to the identity, as is any product of 1-cycles. A 2-cycle is called a transposition, corresponds to interchanging its two entries, and is its own inverse. Any cycle can be decomposed into a (non-unique) product of transpositions. For instance, (1; 2; 3; : : : ; n) = (1; n)(1; n − 1) ::: (1; 3)(1; 2). [This is not unique because the product (a; b)(a; b) = 1 can be inserted any- where.] Any permutation can be uniquely resolved into a product of non-overlapping 1 2 3 : : : n ! cycles | in the permutation , rearrange the columns a1 a2 a3 : : : an 0 ! 1 a1 : : : am−1 ::: 0 so that it starts = (1; a1; : : : ; am−1); then rear- a1 aa1 ::: 1 ::: range the remaining columns in similar cyclic order, beginning with a label 0 not in the set f1; a1; aa1 ; : : : ; am−1g; and continue until all labels are ex- hausted. It is customary to omit 1-cycles from this resolution, which is self-evidently unique. The result of the resolution can be characterised by listing the number of r-cycles, νr, for each value of r from 1 to n. The list fνrg defines the cycle structure of the permutation. A permutation which is resolved into cycles all of the same degree is called a regular permutation and its order is equal to the degree of its cycles. A regular permutation has either only 1-cycles, in which case it is the identity permutation, or no 1-cycles, in which case it leaves no symbol unpermuted. The symmetric group on n objects Sn, of order n!, has many subgroups, each of which is a group of permutations. A group of permutations is regular if all its elements are regular. If all the elements of a group of permutations, except the identity, leave no symbol unpermuted, then the group is regular.
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